Financial Prediction With Constrained Tail Risk

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Financial Prediction With Constrained Tail Risk
Alex Trindade Dept. of Mathematics Statistics,
Texas Tech University Joint work with Stan
Uryasev and Greg Zrazhevsky (Dept. of
Industrial Systems Engineering, U. of
Florida, and U. of Kiev, Ukraine) Alexander
Shapiro (School of Industrial Systems
Engineering, Georgia Tech)
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Controlling/Planning for Failure

• In many applications we wish to control or plan
  for failures of large magnitude
• Bursting of flood levees (hydrology).
• Policy premiums (insurance).
• Risk of loans (banking).
• Failure of materials (engineering).
• Amount of inventory (capacity planning).
• Stock market losses (finance).

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Risk Measures
X continuous random variable with pdf f(x) and
cdf F(x)P(Xx). ? a probability level (0lt ? lt1).
Commonly used in financial risk management

• Value-at-Risk (VaR).

• Conditional Value-at-Risk (CVaR), a.k.a. Expected
  Shortfall

CVaR
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Why is VaR Important?

• In the wake of several spectacular banking
  collapses, international regulators have decreed
  that banks must calculate and report the risk of
  each investment made.
• Insurance industry thinking hard about
  implementing regulation
• Financial management firms increasingly
  interested in assessing their market exposure due
  to increasing volatility (large fluctuations in
  exchange and interest rates exponential growth
  of derivatives market).
• VaR invented by the Riskmetrics group at
  JPMorganChase (a global leader in investment
  banking), c. 1994.
• World leaders in the theory of risk, and risk
  management
• Europe The RiskLab at ETH Zurich (Swiss Federal
  Institute of Technology). Lead by Paul
  Embrechts.
• North America The Quantitative Analytics group
  at Standard Poors (NYC). Lead by Craig
  Friedman.

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Why is CVaR Important?

• Has a number of clear theoretical advantages over
  VaR.
• When optimizing portfolios problem formulations
  with CVaR constraints far more tractable than
  VaR-based counterparts.
• Expect that CVaR-based methods will lead to
  superior description and mitigation of risk.
• Tangible difference would be appreciated by both
  practitioners and regulators.
• More research needs to be done encourage funding
  agencies like NSF to support such work.

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Relationship between VaR and CVaR
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Computation/Estimation of VaR CVaR
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Asymptotics of VaR CVaR

• CVaR Above agrees with Nagaraja (1982) takes
  estimator of CVaR to be sample average of order
  statistics beyond n?

• CVaR Additionally, we are able to show that
  has negative bias in finite samples, and also
  asymptotically where it is O(1/n).

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Linear Regression
Consider observations i1,,n from the linear
regression model
Usually estimate ? via minimization of some
functional of residuals
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Linear Regression - continued

• Quantile Regression (Koenker Bassett, 1978)
  estimate ?-quantile surface of Yx

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CVaR-Constrained Regression
To control over-estimation/prediction, we propose
least absolute (j1 CL1 loss) or squared (j2
CL2 loss) deviations, but with a CVaR constraint
on the residual error distribution (restriction
on solution space B)
where the CVaR of the residuals can be estimated
consistently by
Optimization, over ? and ?, can be carried out
efficiently via linear programming.
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CVaR-Constrained Regression Interpretation
Heuristic interpretation of fitted response
surface,
(CL2 surface passes through )
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Asymptotics of CVaR-Constrained Regression

• If CVaR?(?) lt ? CVaR constraint is not active in
  the limit and the asymptotics of estimates of ?
  are same as in unconstrained case.
• If CVaR?(?) ? estimates of ? are still
  consistent, but their asymptotics are different
  (not asymptotically normal).
• If CVaR?(?) gt ? estimates of ? no longer
  consistent.

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Case Study NDX SPX Indices

• Historical rates of semi-daily returns for NDX
  SPX indices from 07/14/2000 to 06/27/2002. (480
  trading days.)

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Case Study Formation of Data Sets

• For each of the 4 time series, the 480
  consecutive (trading) days of observations were
  split into 10 groups of 48 days each.
• Group k, k1,,10, consists of days k,10k,
  20k,, 470k, i.e. every 10th day to a group.
• Response-Covariate sets YNi, XSi, XNi YSi,
  XSi, XNi .
• For each response-covariate set, use 9 groups
  (470 days) as in-sample data (training set) 1
  group (48 days) as out-of-sample data (test set).
• To each in-sample data set fit polynomial
  regressions with up to 4th order powers in the
  covariates, allowing for interactions.

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Case Study Time Series Plots of one of the
Training Sets
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Case Study Boxplots of Results for Model with
Response YNi

• relative differences between in out-of-sample
  attained CVaR values for all 10 training-test set
  groupings across probability levels
  ?0.50,0.80,0.90,0.95 and constraint values
  ?0,.001,.005,.01,.015,.02. Models are fitted via
  CL1 loss.

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Case Study Contour Plot for Model with Response
YNi
Fitted response surface under CL1 loss for
training set corresponding to test group 1,
with probability level ?0.95, CVaR constraint
?0.01.
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Case Study Contour Plot for Model with Response
YNi (Interpretation)

• NDX SPX overnight returns falling above the
  zero contour line signal when positive returns in
  NDX might be expected overday.
• Resulting 5 worst drops in the NDX overday index
  are on average no lower than -0.01.

zero contour
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Case Study 3D Plot for Model with Response YNi
zero plane
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Case Study Predictions for Models with Response
YNi
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Case Study Residuals for Models with Response
YNi
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Further Research Im Thinking About

• Deviation CVaR Can axiomatize construction of
  measures of deviation (dispersion) dont have to
  be symmetric generalize variance
• 
  (standard deviation, symmetric)
• 
  (deviation CVaR, asymmetric)
• Can now generalize Markov Chebyshev
  Inequalities Cramer-Rao lower bound theory of
  UMVUEs etc. ?
• Deviation CVaR Regression using deviation CVaR?
  in place of LS gives what kind of estimators?
  (Ans same as ?-quantile regression.)
• Optimal Deviation CVaR Prediction generalize
  optimal prediction theory based on MSE.
• How do properties of estimators of VaR? and CVaR?
  compare when we have ?gt? such that VaR? CVaR? ?